The stochastic map filter seeks a non-deterministic
transformation from forecast to filtering distribution. This
transformation is given by a transport map that pushes forward the
joint distribution of state and data at a given observation time
(\(\pi _{X, Y}\) ) to the filtering distribution. We target a
specific transport map derived from the Knothe–Rosenblatt (KR)
rearrangement that pushes forward \(\pi _{X, Y}\)  to a “reference”
distribution with independent components, e.g., a standard normal.
The KR rearrangement is the unique monotone triangular transport map
that defines a deterministic coupling between two distributions
[1]},
and can be estimated within a finite-dimensional function space using
only convex optimization and samples from \(\smash{\pi _{X,Y}}\)  [2]}. These samples are easy to obtain
given a forecast ensemble \((x^i)_{i \le M}\) : it suffices to simulate the
likelihood \(\pi _{Y\vert X}\)  at each particle \(\smash{x^i}\) . If
we restrict the estimator of the rearrangement to be linear, then we
recover the stochastic EnKF [3]}. If we consider
nonlinear parameterizations of the estimator, however, we obtain a
whole new class of nonlinear filtering algorithms, which rely on fast
and robust convex optimization.
